2017
Том 69
№ 6

All Issues

Impulsive differential inclusions involving evolution operators in separable Banach spaces

Benchohra M., Nieto J. J., Ouahab A.

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Abstract

We present some results on the existence of mild solutions and study the topological structure of the sets of solutions for the following first-order impulsive semilinear differential inclusions with initial and boundary conditions: $$y'(t) − A(t)y(t) \in F(t, y(t)) \text{for a.e.} t \in J\ \{t1,..., tm,...\},$$ $$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$ $$y(0) = a$$ and $$y'(t) − A(t)y(t) \in F(t, y(t)) \text{for a.e.} t \in J\ \{t1,..., tm,...\},$$ $$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$ $$Ly = a,$$ where $J = IR_+,\; 0 = t_0 < t_1 <...< t_m < ...;\; (m \in N), \lim_{k→∞} t_k = ∞,\; A(t)$ is the infinitesimal generator of a family of evolution operator $U(t, s)$ on a separable Banach space $E$, and $F$ is a set-valued mapping. The functions $I_k$ characterize the jump of solutions at the impulse points $t_k,\; k = 1,... .$ The mapping $L: P C_b → E$ is a bounded linear operator. We also investigate the compactness of the set of solutions, some regularity properties of the operator solutions, and the absolute retractness.

English version (Springer): Ukrainian Mathematical Journal 64 (2012), no. 7, pp 991-1018.

Citation Example: Benchohra M., Nieto J. J., Ouahab A. Impulsive differential inclusions involving evolution operators in separable Banach spaces // Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 867-891.

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